Series and Sequences

rokoteko asked on #perl6 about using sequences to calculate arbitrarily accurate values. I’m not sure why he thought I was the expert on this, but I do have some ideas, and thought they should be blogged for future reference.

Note that we don’t try to do this exclusively as a sequence; getting 1, 1/3, 1/5, 1/7... is much easier using map, and then we mix that with the sequence of powers of $x using Z*.

So, one nice thing about the sequence operator is that you can easily use it to get all the Rats in the terms of a Taylor series, because once the terms get small enough, they will switch to Nums. So we can say

I use sorted-merge from List::Utils to merge the two sequences of Taylor series terms into one strictly decreasing (in magnitude) sequence. That, in turn, makes it easy to use the triangle-reduce metaoperator to stop summing the terms when they’ve gotten so small they are no longer representable by a Rat.

What is this good for? Well right now, not much. Sure, we’re gotten a fairly accurate Rat version of pi, but we could have gotten that more quickly and accurately by just saying pi.Rat(1e-15).

But once we have working FatRats, this approach will let us get arbitrarily accurate rational approximations to pi. Indeed, it suggests we could have slow but very accurate ways of calculating all sorts of transcendental functions…